Mathematical tools for data mining: set theory, partial orders, combinatorics

Dan A. Simovici, Chabane Djeraba (auth.)1848002009, 9781848002005

The maturing of the field of data mining has brought about an increased level of mathematical sophistication. Such disciplines like topology, combinatorics, partially ordered sets and their associated algebraic structures (lattices and Boolean algebras), and metric spaces are increasingly applied in data mining research. This book presents these mathematical foundations of data mining integrated with applications to provide the reader with a comprehensive reference.

Mathematics is presented in a thorough and rigorous manner offering a detailed explanation of each topic, with applications to data mining such as frequent item sets, clustering, decision trees also being discussed. More than 400 exercises are included and they form an integral part of the material. Some of the exercises are in reality supplemental material and their solutions are included. The reader is assumed to have a knowledge of elementary analysis.

Features and topics:

• Study of functions and relations

• Applications are provided throughout

• Presents graphs and hypergraphs

• Covers partially ordered sets, lattices and Boolean algebras

• Finite partially ordered sets

• Focuses on metric spaces

• Includes combinatorics

• Discusses the theory of the Vapnik-Chervonenkis dimension of collections of sets

This wide-ranging, thoroughly detailed volume is self-contained and intended for researchers and graduate students, and will prove an invaluable reference tool.

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Table of contents :
Front Matter….Pages I-XII
Front Matter….Pages 1-1
Sets, Relations, and Functions….Pages 3-55
Algebras….Pages 57-77
Graphs and Hypergraphs….Pages 79-125
Front Matter….Pages 127-127
Partially Ordered Sets….Pages 129-172
Lattices and Boolean Algebras….Pages 173-224
Topologies and Measures….Pages 225-272
Frequent Item Sets and Association Rules….Pages 273-293
Applications to Databases and Data Mining….Pages 295-332
Rough Sets….Pages 333-348
Front Matter….Pages 349-349
Dissimilarities, Metrics, and Ultrametrics….Pages 351-421
Topologies and Measures on Metric Spaces….Pages 423-458
Dimensions of Metric Spaces….Pages 459-493
Clustering….Pages 495-525
Front Matter….Pages 527-527
Combinatorics….Pages 529-549
The Vapnik-Chervonenkis Dimension….Pages 551-567
Back Matter….Pages 597-616